7.3.1. Assumptions of LOT model
Since the LOT Y-split model is the amended version of Lesmond et al. (1999) (LOT) model, this section starts with discussing the LOT model. According to Lesmond et al. (1999, 2004), the LOT model indirectly infers the transaction costs based on investors’ behaviour. This model assumes that investors rationally assess the (potential) transaction costs which they will face from their investments, such as the bid-ask spread, applicable commissions, price impact costs, taxes, short-sale costs and immediacy costs, before making trading decisions. Investors rationally start trading if the value of investment after transaction costs is profitable.
The LOT model infers monthly transaction costs for each stock through the incidence of zero daily returns over a given period, such as over the last one calendar year used in their study. The basic hypothesis of the LOT model is that a zero return, on average, is observed if the transaction costs threshold is not exceeded. In other words, zero-return is observed if
108 investors do not trade because the trading is not profitable after accounting for the transaction costs.
This implies that zero returns result from the effects of costs on marginal traders, who may be informed or uninformed. For informed traders, if the value of the public-plus-private information is insufficient to exceed the costs of trading, then these marginal investors will either reduce their desired trades or even refrain from trading. Under these circumstances, there will be no price movement from the previous day. For most liquidity traders, if the need for liquidity is sufficiently low and the transaction costs sufficiently high, again they will not trade, which again leads to a zero return. However, some liquidity traders may trade regardless of transaction costs and the resulting returns may be non-zero. The LOT model assumes that the value of their trades is idiosyncratic and over time the average returns resulting from their trades will be zero.
7.3.2. Calculations of the LOT model
This measurement is based on the limited dependent variable (LDV) model to estimate the frequency of zero returns in order to infer the transaction costs of buying 𝛼2 and selling 𝛼1. In the presence of transaction costs, the marginal informed traders will trade only if the value of information exceeds trading costs.
Lesmond et al. (1999) estimate true return 𝑅𝑗𝑡∗ on the basis of the standard market model. 32
Let 𝑅𝑗𝑡∗ be the true returns such that,
𝑅𝑗𝑡∗ = 𝛽
𝑗𝑅𝑚𝑡+ 𝜀𝑗𝑡 1
32
Since the intercept term in the risk model normally captures the misspecification in the market index and not the transaction costs, Lesmond et al. (1999) use the risk model without the intercept term.
109 where 𝛽𝑗 is the sensitivity of stock j to the market return 𝑅𝑚𝑡 on day t, and the error term 𝜀𝑗𝑡
indicates a public information shock on day t. This model assumes that 𝜀𝑗𝑡 is normally distributed with a mean zero and variance 𝜎𝑗2.
Let 𝑅𝑗𝑡 be measured returns from daily prices on stocks, where j denotes the stock j and t is the trading day. The measured daily return will be non-zero only if the true stock returns from market model exceed the transaction costs.
The relationship between the measured return 𝑅𝑗𝑡 and true return 𝑅𝑗𝑡∗ can be shown as
𝑅𝑗𝑡= 𝑅𝑗𝑡∗ − 𝛼1𝑗 𝑖𝑓 𝑅𝑗𝑡∗ < 𝛼1𝑗 (1)
𝑅𝑗𝑡= 0 𝑖𝑓 𝛼1𝑗 < 𝑅𝑗𝑡∗ < 𝛼
2𝑗 (0)
𝑅𝑗𝑡= 𝑅𝑗𝑡∗ − 𝛼
2𝑗 𝑖𝑓 𝑅𝑗𝑡∗ > 𝛼2𝑗 (2)
where 𝛼1𝑗 is the threshold level below which the marginal investors will want to sell if given negative information about stock j. Similarly, 𝛼2𝑗 is the threshold above which the marginal investors will want to buy given positive information about stock j. If the true return is not low enough or high enough to exceed two thresholds levels, the investors will decide not to trade, which causes a zero return. The two threshold levels measure the percentage transaction costs of selling stock j and buying stock j, respectively. Therefore, the proportional round-trip transaction cost of stock j at time t for a competitive marginal investor is the difference between the percentage buying and selling costs,
110 To determine two threshold levels, 𝛼1𝑗 and 𝛼2𝑗, Lesmond et al. (1999) develop the following
maximum likelihood function to estimate the four parameters 𝛼1𝑗, 𝛼2𝑗, 𝛽𝑗, 𝜎𝑗 of the LOT model: L(𝛼1𝑗, 𝛼2𝑗, 𝛽𝑗, 𝜎𝑗|𝑅𝑗𝑡, 𝑅𝑚𝑡) = ∏ 1 𝜎𝑗 1 𝑛 [𝑅𝑗𝑡+ 𝛼1𝑗− 𝛽𝑗𝑅𝑚𝑡 𝜎𝑗 ] × ∏ [𝑁 (𝛼2𝑗 − 𝛽𝑗𝑅𝑚𝑡 𝜎𝑗 ) − 𝑁 ( 𝛼1𝑗 − 𝛽𝑗𝑅𝑚𝑡 𝜎𝑗 )] 0 × ∏ 1 𝜎𝑗 2 𝑛 [𝑅𝑗𝑡+ 𝛼2𝑗− 𝛽𝑗𝑅𝑚𝑡 𝜎𝑗 ] 3 S.T. 𝛼1𝑗 ≤ 0, 𝛼2𝑗 ≥ 0, 𝛽𝑗 ≥ 0, 𝜎𝑗 ≥ 0
where N(.) is the cumulative normal distribution and n(.) is the normal distribution.
By giving the relationship between measured return 𝑅𝑗𝑡 and true return 𝑅𝑗𝑡∗, the LOT model is using an optimisation method to find the maximum measured return 𝑅𝑗𝑡 (return after transaction costs) by estimating four parameters 𝛼1𝑗, 𝛼2𝑗, 𝛽𝑗, 𝜎𝑗. The critical parameters that we are interested in are 𝛼1𝑗 and 𝛼2𝑗, which are potential transaction costs for selling and
buying, respectively. 33
33 Round-trip transaction costs are estimated as a percentage of the price for each stock in each calendar year. As in Liu et al. (2011), stocks have at least 30% nonzero returns in a calendar year are included in the model. The starting values for the estimated parameters 𝛼1𝑗, 𝛼2𝑗, 𝛽𝑗 and 𝜎𝑗 are 0.01, 0.01, 1, and 0.1, respectively. If the procedure fails to converge, the starting values change to 0.1, 0.1, 1, and 0.1, and re-estimate.
111 7.3.3. LOT – Y split model
Goyenko et al. (2009) point out that the definitions of the three regions in the LOT model influence the quality of the estimates. By matching the information of effective and realised spread, and price impact, they show that assigning daily returns into three regions based on stock return itself (𝑅𝑗𝑡) rather than combining with market return (𝑅𝑚𝑡) will be more accurate
for measuring the transaction costs. That is, the returns on the day will be in the region of zero when 𝑅𝑗𝑡 = 0, returns on the day will be in the region of one when 𝑅𝑗𝑡 > 0, and returns on the day will be in the region of 2 when 𝑅𝑗𝑡 < 0, and therefore, the model is named as the
LOT Y-split model.
This study estimates the transaction costs of individual stock following the method employed in Goyenko et al. (2009) and seeks to find whether the two momentum strategies are still profitable across 24 developed markets over last two decades. To my best knowledge, this is the first study to test the two momentum profits after including the transaction costs by using the LOT Y-split measurement model.